Thin Elastic Plates Supported over Small Areas. II: Variational-Asymptotic Models

An asymptotic analysis is performed for thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $\theta_{h}$ of one base with diameter of the same order as the plate thickness $h\ll 1$. A three-dimensional boundary layer in the vicinity of the support $\theta_{h}$ is involved into the asymptotic form which is justified by means of the previously derived weighted inequality of Korn's type provides an error estimate with the bound $ch^{1/2}\left\vert \ln h \right\vert$. Ignoring this boundary layer effect reduces the precision order down to $\left\vert \ln h\right\vert ^{-1/2}$. A two-dimensional variational-asymptotic model of the plate is proposed within the theory of self-adjoint extensions of differential operators. The only characteristics of the boundary layer, namely the elastic logarithmic potential matrix of size $4\times4,$ is involved into the model which however keeps the precision order $h^{1/2}\left\vert \ln h\right\vert$ in certain norms. Several formulations and applications of the model are discussed.